Three Tangent Circles - Problem With Solution
A problem, on three tangent circles, is presented along with detailed solution.
In the figure below, three circles are tangent to each other and to line L. The radius of circle A is equal to 10 cm and the radius of circle B is equal to 8 cm. Find the radius of circle C.
Solution to Problem
Let a, b and c be the radii of the three circles. We first draw the lines AA', BB' and CC' perpendicular to line L. B'C, CA' and BA" are parallel to line L.
Let x = B'C, y=A'C and z = BA". Pythagora's theorem applied to the right triangle BCB' gives
x 2 + BB' 2 = BC 2
Note that BB' = b - c and BC = b + c. The above equation may be written as follows
x 2 + (b - c) 2 = (b + c) 2
Expand the above, group like terms and solve for x
x 2 = 4 b c
x = 2 Square Root( b c )
Use Pythagora's theorem to triangle AA'C to obtain
y 2 + (a - c) 2 = (a + c) 2
Expand, group like terms and solve for y
y 2 = 4 a c
y = 2 Square Root( a c )
Use Pythagora's theorem to triangle AA"B to obtain
z 2 + (a - b) 2 = (a + b) 2
Expand, group and solve for z
z 2 = 4 a b
z = 2 Square Root( a b )
We now use the fact that z = x + y to write
Square Root( a b ) = Square Root( b c )+ Square Root( a c )
Solve the above for c
Square Root( a b ) = Square Root( c )Square Root ( b )+ Square Root( c )Square Root( a )
Square Root( c ) = Square Root( a b ) / [ Square Root ( b )+ Square Root( a ) ]
c = ( a b ) / [ Square Root ( b )+ Square Root( a ) ] 2
Substitute a and b by their values to obtain c
c = 10 * 8 / [ Square Root ( 8 )+ Square Root( 10 ) ] 2
c = 2.2 cm (rounded to 1 decimal place)
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